e to the pi i, a nontraditional take (old version)

The enigmatic equation e^{pi i} = -1 is usually explained using Taylor's formula during a calculus class. This video offers a different perspective, which involves thinking about numbers as actions, and about e^x as something which turns one action into another.

[samlawhorn]

Thank you for leaving this old video up. It's an inspiration to those of us wanting to do our own videos: it tells us your amazing style didn't happen overnight, but rather took some experimentation to get just right.

[nicholasyap]

Before the video:
I came here to understand Euler's Rule
After the video:
Scrolling through comments to see whether anyone else was also lost

[varunmarar]

The amount of knowledge passed down at each instant during this video, is pretty well described by e^(timeline)

[creedfromtheoffice]

What an experience it is to watch this video years later after learning advanced math in college. I remember seeing this years ago and not understanding a thing, now I understood almost everything. You have given me a fantastic new view of mathematics. Learning this stuff in Calculus, Linear Algebra, and Differential Equations was great, but this video really brings it together with an amazing new perspective.

[yousorooo]

First 3 minutes: okay I can keep up with this
Last 3 minutes: wat

[grainfrizz]

Watched this video pretending to myself I understood a thing.

[ExplosiveBrohoof]

I...I think I'll stick with the calculus proof.

[Peterscraps]

Both your pinned comment and the top comments are retrospective and think of this video in lesser terms as if this should meet todays standards. I cannot disagree more, these videos were unique for their time, your skill in translating topics has improved and as a result so has general knowledge.
For people who only have themselves as a test, mistakes are the most memorable teachers.

[ParthaDey]

It's so amazing when you get a completely fresh perspective on things you've already learnt the conventional way

[Epoch]

At about 3 minutes....actually a bit before, you begin to speak very rapidly about complicated concepts. You speak slowly in the beginning and explain the easy stuff in great detail, but then ramble quickly through the material that gives me pause. I enjoy your video as you have a nice speaking voice and good visuals, but I wish you had moved quickly through what was simple and then slowed down for the more complicated issues.

[gabrielherman]

You are great at making things sound simple without making them actually understandable

[teodoranasz]

alright folks, I totally agree 3b1b's level in 2020 is just unmatchable, even by 2015 3b1b himself. but just (pause and ponder) consider his progress. I mean.
also, it's not that this is a better proof for e^(pi*i) or something, but the approach is really unconventional and creative. and since this is such an isolated equation, it's a perfect place to start your channel.
+ notice how back then, he already hat his oustanding and brilliant phrasing, like when he said "the life's ambition of e^x is to transform adders into mulitpliers" - that's just talent. this doesn't have to be your (or Grants) favorite way to think about e^(pi*i), but it's a different approach so it's inherently worth considering.

[muesk]

This would work so much better if you had actual numbers on there. You're rotating.. ok great.. hard to see what is actually happening when there are no numbers!

[huzzzzzzahh]

I think this is really an explanation for mathematicians more than for lay people; to really follow it, one needs to be comfortable with reframing things in an abstract way, defining functions by functional equations, and choosing things based on naturality. To a mathematician, this makes perfect sense: view the real numbers as their actions on the geometric line. What could be more reasonable? It is a familiar thing to do, also, with many examples in mathematics of great success with this abstract (dare I say relative (a la Grothendieck)) point of view. And then all you have to do is choose whatever is most natural, which most mathematicians will 'naturally' do, and everything falls out beautifully. But to many it seems like hand waving, even though it's not; it's definition waving, which I have no problem with at all!

[mihailazar]

These older videos are way more artistic in animation but they're 2fast, 2 quick 4 anyone to understand
It's a good thing he started talking slower as time went on and those moments where they stop by and have the pi figures talk is a great time to let the facts sink in and not rush though the entire video at break-neck speed

[Rangvald]

I've struggled with math my entire life, it's one of the only disciplines that hasn't just come naturally to me. Your videos make way more sense to me than anything that my math teachers could have come up with growing up.

[nicolasyan]

I really think you should revisit these earlier videos, especially this one and the one you link to at the end of it, with your newer, clearer style. Your later videos have animations that are even more beautiful and with narration that is easier to follow.

[connorking]

WOW redefining what numbers are intrinsically. This is what I've always been curious about but never actually been able to ask about because when I ask my teachers "how can I think about multiplication fundamentally" they look at me like I'm stupid because I don't know what it is.

This is it man, thank you so

[pontifex]

Oh, it's all clear to me now. How could I not have seen this before?

[johncgibson]

OMG. All engineering students and professor and engineers should watch this to get the real sense of e. This episode should have 1 billion views.